Let a, b, c be non-zero real numbers such that ; _0^1 (1 + cos^8x… - Vidyadip

MCQ

Let $a, b, c$ be non-zero real numbers such that ; $\int\limits_0^1 {} (1 + cos^8x) (ax^2 + bx + c) dx$ $= \int\limits_0^2 {} (1 + cos^8x) (ax^2 + bx + c) dx$ , then the quadratic equation $ax^2 + bx + c = 0$ has :

A
no root in $(0, 2)$
✓
atleast one root in $(0, 2)$
C
a double root in $(0, 2)$
D
none

✓

Answer

Correct option: B.

atleast one root in $(0, 2)$

b
$\int\limits_0^1 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $

$=\int\limits_0^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $

$=\int\limits_0^1 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} \,\, + \,\,\int\limits_1^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} $
Hence $\,\int\limits_1^2 {(1 + {{\cos }^8}x)\,\,f(x)\,\,dx} = 0$
$\Rightarrow (1+cos^8x) f(x) = 0$ at least once in $(1,2)$
but $1 + \cos^8x \neq 0$
$\Rightarrow f(x) = ax^2 + bx + c$ vanishes at least once in $(1,2)$

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